Problem 86 - Elementary Algebra

Question

Simplify. $$ -(-(-1)) $$

Step-by-Step Solution

Answer

The expression simplifies to -1.

Step 1: Understand the Problem

We have the expression \(-(-(-1))\) that we need to simplify. It consists of nested negative signs. Our goal is to determine the resultant value by simplifying these negative signs.

Step 2: Simplify the Inner Negation

Start by simplifying the innermost part of the expression: \(-1\). The negation of \(-1\) is simply \(1\) because negating a negative number results in a positive number. This changes our expression to \(-(-(1))\).

Step 3: Simplify the Outer Negation

Next, simplify the remaining expression \(-1\). The negation of a positive number \(1\) is \(-1\). Therefore, the entire expression simplifies to \(-1\).

Step 4: Final Simplified Expression

The expression \(-(-(-1))\) simplifies to \(-1\). You can think of the three negatives as pairs of negatives making positives, leaving one negative sign.

Key Concepts

Understanding Negation

Negation is an essential concept in mathematics, particularly when dealing with integers. To negate a number means to reverse its sign. For example, negating a positive number results in a negative number, and vice versa.
Consider the number -1. To negate -1, we reverse its sign, resulting in 1. Essentially, if you see a negative sign in front of a number, you can think of it as flipping the number's sign.

• Negative of a positive number: \(- (+n) = -n\)
• Negative of a negative number: \(- (-n) = n\)

This simple rule is the foundation when we work with negative integers. Proper understanding helps when simplifying expressions involving multiple negations.

Mastering Simplification

Simplification in mathematics is all about making an expression easier to work with. For expressions involving numbers, simplification often involves combining like terms and performing operations to arrive at a simpler form.
When dealing with integers and negations, simplification might involve reducing the number of negations by combining or resolving them, as was the case with the expression \(-(-(-1))\). Here's how it works:

• Simplify inner expressions first. Always look at the innermost part of the expression, simplify it, and then proceed outward.
• Remember the effect of negation. Each pair of negatives results in a positive.

By following these steps, you reduce the complexity of the expression, making it easier to understand and work with. For example, in the expression \(-(-(-1))\), resolving it step-by-step, we find it simplifies to \(-1\).

Exploring Nested Expressions

Nested expressions can initially seem intimidating because they involve expressions within expressions. With careful attention, you can simplify these expressions step by step.
To tackle nested expressions effectively:

• Work from the inside out. Start solving the innermost expression and gradually move to the outer layers. This approach helps prevent errors and makes simplifying more manageable.
• Apply known rules, like those for negation and simplification, as you work through the layers.
• Be mindful of each operation's effect, particularly with integers and negative signs, to ensure accuracy.

In our example of simplifying \(-(-(-1))\), each layer of negation was resolved systematically: negating the innermost \(-1\) to \(1\), then addressing the outer layers to reach the final simplified result. Recognizing and managing nested elements is critical in mastering integer operations and other complex mathematical tasks.